Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ssbbi Structured version   Visualization version   GIF version

Theorem bj-ssbbi 32991
Description: Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2493. (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-ssbbi (∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓))

Proof of Theorem bj-ssbbi
StepHypRef Expression
1 biimp 206 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21alimi 1906 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓))
3 bj-ssbim 32990 . . 3 (∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))
42, 3syl 17 . 2 (∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))
5 biimpr 211 . . . 4 ((𝜑𝜓) → (𝜓𝜑))
65alimi 1906 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜓𝜑))
7 bj-ssbim 32990 . . 3 (∀𝑥(𝜓𝜑) → ([𝑡/𝑥]b𝜓 → [𝑡/𝑥]b𝜑))
86, 7syl 17 . 2 (∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜓 → [𝑡/𝑥]b𝜑))
94, 8impbid 203 1 (∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650  [wssb 32988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005
This theorem depends on definitions:  df-bi 198  df-ssb 32989
This theorem is referenced by:  bj-ssbbii  32993
  Copyright terms: Public domain W3C validator